No, you don't. It's fine without the dots.

No, you don't have to, but you may. Both PE and P.E. are understood and accepted usage. "Phys Ed" is also a common abbreviation.

In both British and American English, if you are using initial (i.e. first) letters to represent words, you don’t normally need to put a full stop after them. Example BBC, UK, NATO In American English, it is common to use a full stop as an alternative style for certain** abbreviations. Example USA and U.S.A. both are right abbreviation for "United State of America" PE is abbreviation of Physical Education. PE is more formal and correct without dots(.)

Yes, there is a name called abbreviation.